Extremes of multidimensional stationary Gaussian random fields

• Autorzy: Soja-Kukieła N.

Identyfikatory

DOI

10.19195/0208-4147.38.1.10

• Języki publikacji: EN

Abstrakty

EN

Let {X(t) : t = (t₁, t₂,…, t_d) ϵ [0, ∞)^d} be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function r satisfying r(t) < 1 for every t ≠ 0 and r(t) = 1 – Σ^d_i=1 |t_i|^α_i + o (Σ^d_i=1 |t_i|^α_i), as t → 0, with some α₁, α₂,…, α_d ϵ (0, 2]. The main result of this contribution is the description of the asymptotic behaviour of P (sup{X(t) : t ϵ J^x_m} ≤ u), as u → ∞, for some Jordan-measurable sets J^x_m of volume proportional to P (sup{X(t) : t ϵ [0, 1]^d} > u)⁻¹ (1 + o(1)).

Słowa kluczowe

EN

Gaussian random field; supremum; limit theorem; asymptotics; Berman condition; strong dependence

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2018
• Tom: Vol. 38, Fasc. 1
• Strony: 191--207
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Soja-Kukieła N.

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [1] M. Arendarczyk and K. Dębicki, Exact asymptotics of supremum of a stationary Gaussian process over a random interval, Statist. Probab. Lett. 82 (2012), pp. 645-652.
• [2] K. Dębicki, E. Hashorva, and N. Soja-Kukieła, Extremes of stationary Gaussian random fields, preprint, available at http://arxiv.org/abs/1312.2863, 2013.
• [3] K. Dębicki, E. Hashorva, and N. Soja-Kukieła, Extremes of stationary Gaussian random fields, J. Appl. Probab. 52 (2015), pp. 55-67.
• [4] M. R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer Ser. Statist., Springer, New York 1983.
• [5] J. III Pickands, Asymptotic properties of maximum in a stationary Gaussian process, Trans. Amer. Math. Soc. 145 (1969), pp. 75-86.
• [6] V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Transl. Math. Monogr., Vol. 148, American Mathematical Society, Providence 1996.
• [7] Z. Tan and E. Hashorva, Exact tail asymptotics of the supremum of a strongly dependent Gaussian process over a random interval, Lith. Math. J. 53 (2013), pp. 91-102.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).